Exactly determining the expansion coefficients in the recovery of space-dependent pollutant source

In this paper, after deriving certain adjoint eigenfunctions as test functions, as well as the bases of unknown pollutant source, the numerical algorithm based on the weak-form integral equation method is developed, of which the expansion coefficients can be found exactly in closed-form, without resorting to numerical solution. The closed-form expansion coefficients method is effective for the inverse pollutant source problem (IPSP), which can recover the unknown pollutant source by means of an extra final time condition. Numerical examples of the IPSP demonstrate that the closed-form expansion coefficients method is stable against large noises up to 10%-50%.